Distribution function approach to redshift space distortions

TL;DR

This work introduces a phase-space distribution-function framework for redshift-space distortions (RSD) that expresses the redshift-space density as a sum over mass-weighted velocity moments and organizes these moments via a helicity decomposition. The redshift-space power spectrum $P^{ss}(k)$ is derived as a double sum over velocity-moment correlators $P_{LL'}(k)$ with angular dependence governed by associated Legendre polynomials $P_l^m(\mu)$, ensuring rotational invariance and a systematic expansion in $k_\parallel/\mathcal{H}$. The leading μ^2 term arises from the density–scalar momentum cross-correlation (tied to $dP_{00}/d\tau$), while μ^4 receives seven distinct, second-order or higher contributions from momentum, energy density, and anisotropic-stress correlations; the framework also accounts for FoG-like damping through $P_{02}$ terms and clarifies the role of vorticity in momentum correlations. The approach remains valid on large scales and in multi-stream scenarios, highlights the impact of bias on velocity-moment correlators for galaxies, and provides a physically transparent, systematically improvable alternative to single-parameter FoG models for modeling RSD in galaxy surveys.

Abstract

We develop a phase space distribution function approach to redshift space distortions (RSD), in which the redshift space density can be written as a sum over velocity moments of the distribution function. These moments are density weighted and their lowest orders are density, momentum density, and stress energy density. The series expansion is convergent on large scales. We perform an expansion of these velocity moments into helicity modes, which are eigenmodes under rotation around the axis of Fourier mode direction, generalizing the scalar, vector, tensor decomposition of perturbations to an arbitrary order. We show that only equal helicity moments correlate and derive the angular dependence of the individual contributions to the redshift space power spectrum in terms of angle mu between wave vector and line of sight. We show that the dominant term of mu^2 dependence on large scales is the cross-correlation between the density and scalar part of momentum density, which can be related to the time derivative of the matter power spectrum. Additional terms contributing and dominating on small scales are the vector part of momentum density-momentum density correlations, the energy density-density correlations, and the scalar part of anisotropic stress density-density correlations. Similarly, we identify 7 terms contributing to mu^4 dependence. Some of the advantages of the distribution function approach are that the series expansion converges on large scales and remains valid in multi-stream situations. We finish with a brief discussion of implications for RSD in galaxies relative to dark matter, highlighting the issue of scale dependent bias of velocity moments correlators.